In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also capacity, see ), which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different classes of fuzzy measures including plausibility/belief measures, possibility/necessity measures, and probability measures, which are a subset of classical measures. == Definitions == Let X {\displaystyle \mathbf {X} } be a universe of discourse, C {\displaystyle {\mathcal {C}}} be a class of subsets of X {\displaystyle \mathbf {X} } , and E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} . A function g : C → R {\displaystyle g:{\mathcal {C}}\to \mathbb {R} } where ∅ ∈ C ⇒ g ( ∅ ) = 0 {\displaystyle \emptyset \in {\mathcal {C}}\Rightarrow g(\emptyset )=0} E ⊆ F ⇒ g ( E ) ≤ g ( F ) {\displaystyle E\subseteq F\Rightarrow g(E)\leq g(F)} is called a fuzzy measure. A fuzzy measure is called normalized or regular if g ( X ) = 1 {\displaystyle g(\mathbf {X} )=1} . == Properties of fuzzy measures == A fuzzy measure is: additive if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} such that E ∩ F = ∅ {\displaystyle E\cap F=\emptyset } , we have g ( E ∪ F ) = g ( E ) + g ( F ) . {\displaystyle g(E\cup F)=g(E)+g(F).} ; supermodular if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} , we have g ( E ∪ F ) + g ( E ∩ F ) ≥ g ( E ) + g ( F ) {\displaystyle g(E\cup F)+g(E\cap F)\geq g(E)+g(F)} ; submodular if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} , we have g ( E ∪ F ) + g ( E ∩ F ) ≤ g ( E ) + g ( F ) {\displaystyle g(E\cup F)+g(E\cap F)\leq g(E)+g(F)} ; superadditive if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} such that E ∩ F = ∅ {\displaystyle E\cap F=\emptyset } , we have g ( E ∪ F ) ≥ g ( E ) + g ( F ) {\displaystyle g(E\cup F)\geq g(E)+g(F)} ; subadditive if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} such that E ∩ F = ∅ {\displaystyle E\cap F=\emptyset } , we have g ( E ∪ F ) ≤ g ( E ) + g ( F ) {\displaystyle g(E\cup F)\leq g(E)+g(F)} ; symmetric if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} , we have | E | = | F | {\displaystyle |E|=|F|} implies g ( E ) = g ( F ) {\displaystyle g(E)=g(F)} ; Boolean if for any E ∈ C {\displaystyle E\in {\mathcal {C}}} , we have g ( E ) = 0 {\displaystyle g(E)=0} or g ( E ) = 1 {\displaystyle g(E)=1} . Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integral or Choquet integral, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging (OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral. == Möbius representation == Let g be a fuzzy measure. The Möbius representation of g is given by the set function M, where for every E , F ⊆ X {\displaystyle E,F\subseteq X} , M ( E ) = ∑ F ⊆ E ( − 1 ) | E ∖ F | g ( F ) . {\displaystyle M(E)=\sum _{F\subseteq E}(-1)^{|E\backslash F|}g(F).} The equivalent axioms in Möbius representation are: M ( ∅ ) = 0 {\displaystyle M(\emptyset )=0} . ∑ F ⊆ E | i ∈ F M ( F ) ≥ 0 {\displaystyle \sum _{F\subseteq E|i\in F}M(F)\geq 0} , for all E ⊆ X {\displaystyle E\subseteq \mathbf {X} } and all i ∈ E {\displaystyle i\in E} A fuzzy measure in Möbius representation M is called normalized if ∑ E ⊆ X M ( E ) = 1. {\displaystyle \sum _{E\subseteq \mathbf {X} }M(E)=1.} Möbius representation can be used to give an indication of which subsets of X interact with one another. For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons. The fuzzy measure g in standard representation can be recovered from the Möbius form using the Zeta transform: g ( E ) = ∑ F ⊆ E M ( F ) , ∀ E ⊆ X . {\displaystyle g(E)=\sum _{F\subseteq E}M(F),\forall E\subseteq \mathbf {X} .} == Simplification assumptions for fuzzy measures == Fuzzy measures are defined on a semiring of sets or monotone class, which may be as granular as the power set of X, and even in discrete cases the number of variables can be as large as 2|X|. For this reason, in the context of multi-criteria decision analysis and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure is additive, it will hold that g ( E ) = ∑ i ∈ E g ( { i } ) {\displaystyle g(E)=\sum _{i\in E}g(\{i\})} and the values of the fuzzy measure can be evaluated from the values on X. Similarly, a symmetric fuzzy measure is defined uniquely by |X| values. Two important fuzzy measures that can be used are the Sugeno- or λ {\displaystyle \lambda } -fuzzy measure and k-additive measures, introduced by Sugeno and Grabisch respectively. === Sugeno λ-measure === The Sugeno λ {\displaystyle \lambda } -measure is a special case of fuzzy measures defined iteratively. It has the following definition: ==== Definition ==== Let X = { x 1 , … , x n } {\displaystyle \mathbf {X} =\left\lbrace x_{1},\dots ,x_{n}\right\rbrace } be a finite set and let λ ∈ ( − 1 , + ∞ ) {\displaystyle \lambda \in (-1,+\infty )} . A Sugeno λ {\displaystyle \lambda } -measure is a function g : 2 X → [ 0 , 1 ] {\displaystyle g:2^{X}\to [0,1]} such that g ( X ) = 1 {\displaystyle g(X)=1} . if A , B ⊆ X {\displaystyle A,B\subseteq \mathbf {X} } (alternatively A , B ∈ 2 X {\displaystyle A,B\in 2^{\mathbf {X} }} ) with A ∩ B = ∅ {\displaystyle A\cap B=\emptyset } then g ( A ∪ B ) = g ( A ) + g ( B ) + λ g ( A ) g ( B ) {\displaystyle g(A\cup B)=g(A)+g(B)+\lambda g(A)g(B)} . As a convention, the value of g at a singleton set { x i } {\displaystyle \left\lbrace x_{i}\right\rbrace } is called a density and is denoted by g i = g ( { x i } ) {\displaystyle g_{i}=g(\left\lbrace x_{i}\right\rbrace )} . In addition, we have that λ {\displaystyle \lambda } satisfies the property λ + 1 = ∏ i = 1 n ( 1 + λ g i ) {\displaystyle \lambda +1=\prod _{i=1}^{n}(1+\lambda g_{i})} . Tahani and Keller as well as Wang and Klir have shown that once the densities are known, it is possible to use the previous polynomial to obtain the values of λ {\displaystyle \lambda } uniquely. === k-additive fuzzy measure === The k-additive fuzzy measure limits the interaction between the subsets E ⊆ X {\displaystyle E\subseteq X} to size | E | = k {\displaystyle |E|=k} . This drastically reduces the number of variables needed to define the fuzzy measure, and as k can be anything from 1 (in which case the fuzzy measure is additive) to X, it allows for a compromise between modelling ability and simplicity. ==== Definition ==== A discrete fuzzy measure g on a set X is called k-additive ( 1 ≤ k ≤ | X | {\displaystyle 1\leq k\leq |\mathbf {X} |} ) if its Möbius representation verifies M ( E ) = 0 {\displaystyle M(E)=0} , whenever | E | > k {\displaystyle |E|>k} for any E ⊆ X {\displaystyle E\subseteq \mathbf {X} } , and there exists a subset F with k elements such that M ( F ) ≠ 0 {\displaystyle M(F)\neq 0} . == Shapley and interaction indices == In game theory, the Shapley value or Shapley index is used to indicate the weight of a game. Shapley values can be calculated for fuzzy measures in order to give some indication of the importance of each singleton. In the case of additive fuzzy measures, the Shapley value will be the same as each singleton. For a given fuzzy measure g, and | X | = n {\displaystyle |\mathbf {X} |=n} , the Shapley index for every i , … , n ∈ X {\displaystyle i,\dots ,n\in X} is: ϕ ( i ) = ∑ E ⊆ X ∖ { i } ( n − | E | − 1 ) ! | E | ! n ! [ g ( E ∪ { i } ) − g ( E ) ] . {\displaystyle \phi (i)=\sum _{E\subseteq \mathbf {X} \backslash \{i\}}{\frac {(n-|E|-1)!|E|!}{n!}}[g(E\cup \{i\})-g(E)].} The Shapley value is the vector ϕ ( g ) = ( ψ ( 1 ) , … , ψ ( n ) ) . {\displaystyle \mathbf {\phi } (g)=(\psi (1),\dots ,\psi (n)).}
Shaded Picture System
The Shaded Picture System was a 3D raster computer display processor introduced by Evans & Sutherland in October 1973. The Shaded Picture System was the first general-purpose, commercially available raster computer graphics display processor capable of real-time, shaded 3D graphics. It could only display black and white graphics at a resolution of 256 by 256. It was extremely expensive, and very few units were ever sold. == History == The principles of shaded, hidden-line true 3D graphics were pioneered at the University of Utah in 1967. However, this algorithm was slow and would take several minutes to produce an image. In 1970, Gary Watkins developed a FORTRAN simulator of a faster algorithm that would theoretically generate shaded 3D images in real-time, "if implemented in suitable hardware". The simulator itself was still not capable of real-time shaded 3D image rendering. Evans & Sutherland developed a functional prototype of this "suitable hardware", which was later sold as the Shaded Picture System in 1973. About a year earlier in 1972, Evans & Sutherland sold the first and only CT1 to Case Western Reserve University. The CT1, or Continuous Tone 1, was a specialized image generator, not meant as a marketable or mass-produced product. At the time, the CT1, along with G.E./NASA's upgraded Electronic Scene Generator from 1971, would have been the only real-time raster graphics systems sold to customers comparable to the Shaded Picture System, although both the CT1 and Electronic Scene Generator were intentionally produced as one-off products and specialized for the needs of their customers. The Shaded Picture System, in contrast, was intentionally marketed.In early 1975, Evans & Sutherland demonstrated a random-access video frame buffer using relatively low-cost semiconductor memory, which was much more capable than the Shaded Picture System. When interfaced with a (non-shaded) E&S Picture System, the frame buffer had a resolution of 512 by 512 in grayscale and partial color capabilities. By the end of 1975, this frame buffer was commercially available.
The Best Free AI Text-to-image Tool for Beginners
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Markov partition
A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics. By using a Markov partition, the system can be made to resemble a discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov shift. The appellation 'Markov' is appropriate because the resulting dynamics of the system obeys the Markov property. The Markov partition thus allows standard techniques from symbolic dynamics to be applied, including the computation of expectation values, correlations, topological entropy, topological zeta functions, Fredholm determinants and the like. == Motivation == Let ( M , φ ) {\displaystyle (M,\varphi )} be a discrete dynamical system. A basic method of studying its dynamics is to find a symbolic representation: a faithful encoding of the points of M {\displaystyle M} by sequences of symbols such that the map φ {\displaystyle \varphi } becomes the shift map. Suppose that M {\displaystyle M} has been divided into a number of pieces E 1 , E 2 , … , E r {\displaystyle E_{1},E_{2},\ldots ,E_{r}} which are thought to be as small and localized, with virtually no overlaps. The behavior of a point x {\displaystyle x} under the iterates of φ {\displaystyle \varphi } can be tracked by recording, for each n {\displaystyle n} , the part E i {\displaystyle E_{i}} which contains φ n ( x ) {\displaystyle \varphi ^{n}(x)} . This results in an infinite sequence on the alphabet { 1 , 2 , … , r } {\displaystyle \{1,2,\ldots ,r\}} which encodes the point. In general, this encoding may be imprecise (the same sequence may represent many different points) and the set of sequences which arise in this way may be difficult to describe. Under certain conditions, which are made explicit in the rigorous definition of a Markov partition, the assignment of the sequence to a point of M {\displaystyle M} becomes an almost one-to-one map whose image is a symbolic dynamical system of a special kind called a shift of finite type. In this case, the symbolic representation is a powerful tool for investigating the properties of the dynamical system ( M , φ ) {\displaystyle (M,\varphi )} . == Formal definition == A Markov partition is a finite cover of the invariant set of the manifold by a set of curvilinear rectangles { E 1 , E 2 , … , E r } {\displaystyle \{E_{1},E_{2},\ldots ,E_{r}\}} such that For any pair of points x , y ∈ E i {\displaystyle x,y\in E_{i}} , that W s ( x ) ∩ W u ( y ) ∈ E i {\displaystyle W_{s}(x)\cap W_{u}(y)\in E_{i}} Int E i ∩ Int E j = ∅ {\displaystyle \operatorname {Int} E_{i}\cap \operatorname {Int} E_{j}=\emptyset } for i ≠ j {\displaystyle i\neq j} If x ∈ Int E i {\displaystyle x\in \operatorname {Int} E_{i}} and φ ( x ) ∈ Int E j {\displaystyle \varphi (x)\in \operatorname {Int} E_{j}} , then φ [ W u ( x ) ∩ E i ] ⊃ W u ( φ x ) ∩ E j {\displaystyle \varphi \left[W_{u}(x)\cap E_{i}\right]\supset W_{u}(\varphi x)\cap E_{j}} φ [ W s ( x ) ∩ E i ] ⊂ W s ( φ x ) ∩ E j {\displaystyle \varphi \left[W_{s}(x)\cap E_{i}\right]\subset W_{s}(\varphi x)\cap E_{j}} Here, W u ( x ) {\displaystyle W_{u}(x)} and W s ( x ) {\displaystyle W_{s}(x)} are the unstable and stable manifolds of x, respectively, and Int E i {\displaystyle \operatorname {Int} E_{i}} simply denotes the interior of E i {\displaystyle E_{i}} . These last two conditions can be understood as a statement of the Markov property for the symbolic dynamics; that is, the movement of a trajectory from one open cover to the next is determined only by the most recent cover, and not the history of the system. It is this property of the covering that merits the 'Markov' appellation. The resulting dynamics is that of a Markov shift; that this is indeed the case is due to theorems by Yakov Sinai (1968) and Rufus Bowen (1975), thus putting symbolic dynamics on a firm footing. Variants of the definition are found, corresponding to conditions on the geometry of the pieces E i {\displaystyle E_{i}} . == Examples == Markov partitions have been constructed in several situations. Anosov diffeomorphisms of the torus. Dynamical billiards, in which case the covering is countable. Markov partitions make homoclinic and heteroclinic orbits particularly easy to describe. The system ( [ 0 , 1 ) , x ↦ 2 x m o d 1 ) {\displaystyle ([0,1),x\mapsto 2x\ mod\ 1)} has the Markov partition E 0 = ( 0 , 1 / 2 ) , E 1 = ( 1 / 2 , 1 ) {\displaystyle E_{0}=(0,1/2),E_{1}=(1/2,1)} , and in this case the symbolic representation of a real number in [ 0 , 1 ) {\displaystyle [0,1)} is its binary expansion. For example: x ∈ E 0 , T x ∈ E 1 , T 2 x ∈ E 1 , T 3 x ∈ E 1 , T 4 x ∈ E 0 ⇒ x = ( 0.01110... ) 2 {\displaystyle x\in E_{0},Tx\in E_{1},T^{2}x\in E_{1},T^{3}x\in E_{1},T^{4}x\in E_{0}\Rightarrow x=(0.01110...)_{2}} . The assignment of points of [ 0 , 1 ) {\displaystyle [0,1)} to their sequences in the Markov partition is well defined except on the dyadic rationals - morally speaking, this is because ( 0.01111 … ) 2 = ( 0.10000 … ) 2 {\displaystyle (0.01111\dots )_{2}=(0.10000\dots )_{2}} , in the same way as 1 = 0.999 … {\displaystyle 1=0.999\dots } in decimal expansions.
Ronald J. Williams
Ronald James Williams (1945 – February 16, 2024) was an American mathematician and computer scientist who spent the majority of his career at Northeastern University. He is considered one of the pioneers of neural networks. In 1986, he co-authored the seminal paper in Nature on the backpropagation algorithm along with David Rumelhart and Geoffrey Hinton, which triggered a boom in neural network research. == Education and career == Williams was born in Southern California. He studied at California Institute of Technology as a undergraduate student and received a B.S. in mathematics there in 1966. He received his M.A. and Ph.D. in mathematics, both at University of California, San Diego (UCSD) in 1972 and 1975, respectively. His Ph.D. thesis was supervised by Donald Werner Anderson. He worked for a defense contractor for some time after graduation. From 1983 to 1986, Williams was a member of the Parallel Distributed Processing research group headed by David Rumelhart at the Institute for Cognitive Science at UCSD. In 1986, Williams accepted a professorship in computer science at Northeastern University in Boston, where he remained afterwards. In addition to the backpropagation paper, Williams made fundamental contributions to the fields of recurrent neural networks, where he, along with David Zipser, invented the teacher forcing algorithm and made important contributions to backpropagation through time. In reinforcement learning, Williams introduced the REINFORCE algorithm in 1992, which became the first policy gradient method. Besides his works on neural networks, Williams, together with Wenxu Tong and Mary Jo Ondrechen, developed Partial Order Optimum Likelihood (POOL), a machine learning method used in the prediction of active amino acids in protein structures. POOL is a maximum likelihood method with a monotonicity constraint and is a general predictor of properties that depend monotonically on the input features.
TalkBack
TalkBack is an accessibility service for the Android operating system that helps blind and visually impaired users to interact with their devices. It uses spoken words, vibration and other audible feedback to allow the user to know what is happening on the screen allowing the user to better interact with their device. The service is pre-installed on many Android devices, and it became part of the Android Accessibility Suite in 2017. According to the Google Play Store, the Android Accessibility Suite has been downloaded over five billion times, including devices that have the suite preinstalled. == Open-source == Google releases the source code of TalkBack with some releases of the accessibility service to GitHub, with the latest of these changes being from May 6, 2021. The source for these versions of Google TalkBack have been released under the Apache License version 2.0. == Release history ==
Baidu Fanyi
Baidu Fanyi is a service for translating text paragraphs and web pages provided by Baidu. In 2015, Baidu Translation won the second prize of China's National Science and Technology Progress Award. == Supported languages == Baidu translate has some languages that are missing from Google Translate, such as Cornish, albeit some of them are poor quality. As of June 2026, translation is available in 201 languages: